function

exp

(A~/kT) has a sharp peak at small AT is not satisfied.

It can be seen from Figs. 1 and 2 that at a small amount of supercooling the departure of 12 from unity is appreciable, though at the supercooling usually encountered in nozzles and jets ( ~ 5 ~ for water vapor), the departure of 12 from unity is slight. Analysis of the results of the calculations also shows that with increasing dT/dt or decreasing u and saturation temperature of the flow the influence of the nonstationarity of the process of formation of the nucleating centers on their rate of formation increases. This is due to the fact that for given AT with decreasing T H and y and increasing dT/dt the number of collisions between molecules and the nucleating center is decreased. LITERATURE CITED i . A. Kantrowitz, "Nucleation i n very r a p i d vapor expansions," J . Chem. Phys., 19, No. 9 (1951). 2. R. F . P r o b s t e i n , "Time lag in the self-nucleation of a supersaturated v a p o r , " J . Chem. P h y s . , 1 9 , No. 5 ( 1 9 5 1 ) . 3. R. B e c k e t a n d W. D S r i n g , " K i n e t i s c h e Behandlung der Keimbildung in Hbersfittigen D~impfen," Ann. P h y s i k . , 2 4 , No. 8 ( 1 9 3 5 ) . 4 . Ya. I . F r e n k e l , Kinetic Theory of Liquids, Clarendon Press, Oxford (1961). 5. V. M. P a s k o n o v , S t a n d a r d P r o g r a m f o r S o l v i n g B o u n d a r y - L a y e r Problems. Numerical Methods i n Gas D y n a m i c s [ i n R u s s i a n ] ( S e r i e s P r o d u c e d by t h e C o m p u t a t i o n a l C e n t e r a t t h e Moscow State University, No. 2 ) , I z d - v o MGU, Moscow ( 1 9 6 3 ) .

NUMERICAL SIMULATION OF TWO-PI~SE FLOW TIIROUGH POROUS N~DIA WITH ALLOWANCE P.

V.

Indel'man

FOR THE END EFFECT a n d R. M. K a t s

UDC 5 3 2 . 5 4 6

To s i m u l a t e the "end effect" [ 1 ] , o n e u s e s t h e m o d e l o f t w o - p h a s e f l o w [2] a u g m e n t e d by b o u n d a r y c o n d i t i o n s at t h e end a f t e r t h e a p p r o a c h o f t h e w e t t i n g p h a s e t o t h e end s e c tion. The existence of the end effect is explained by t h e d i s c o n t i n u i t y in the properties b e t w e e n t h e o u t e r and i n n e r s i d e s o f t h e end s e c t i o n . In particular, the capillary pressure, which has a finite value within the stratum, b e c o m e s e q u a l t o z e r o on t h e t r a n s i t i o n to the outer region. Therefore, the capillary forces tend to keep the wetting phase within the stratum, w h i c h l e a d s t o an i n c r e a s e in the fraction o f i t s v o l u m e n e a r t h e end s e c t i o n . To d i s p l a c e the wetting phase from the stratum, one must produce a definite pressure gradient. To t a k e i n t o a c c o u n t t h i s p h e n o m enon in the Rapoport--Leas model it is assumed that the wetting phase flows through the end section o n l y when i t h a s r e a c h e d t h e r e i t s m a x i m a l s a t u r a t i o n [3]. The basis of the proposed model is the physical mechanism of displacement of the nonw e t t i n g p h a s e by t h e w e t t i n g p h a s e f r o m a c a p i l l a r y . We g i v e h e r e t h e r e s u l t s of numerical calculations of concurrent-countercurrent imbibition an d d i s p l a c e m e n t problems. We show the influence of the end effect on the nature of the displacement as a f u n c t i o n of the flow velocity. We h a v e f o u n d t h a t t h e n o n w e t t i n g p h a s e d o e s n o t f l o w o u t c o n t i n u o u s l y after the appearance of the wetting fluid at the end, but in a pulsating manner. 1. We c o n s i d e r flow in a region D of rectangular shape with edges 1 and b containing t h e w e t t i n g p h a s e ( w a t e r ) and t h e n o n w e t t i n g p h a s e ( o i l ) , which fill t h e r e g i o n s D1 a n d D2, r e s p e c t i v e l y . We d e n o t e t h e i n t e r f a c e o f t h e two l i q u i d s b y F. We t a k e t h e a x e s o f the coordinates x and y a l o n g t h e e d g e s 1 and b o f t h e r e c t a n g l e . We r e s t r i c t ourselves to considering the class of problems with given pressure difference and conditions of n o - f l o w a t t wo o p p o s i t e s i d e s o f t h e r e g i o n D. We c a l l t h e s e c t i o n x = 0 the inflow and the section x = 1 the outflow. 0il is delivered to the section x = 1. Once t h e w a t e r h a s r e a c h e d t h e o u t f l o w , F c a n be d i v i d e d i n t o two p a r t s : t h e b o u n d a r y F 0 t h a t i s w i t h i n D, a n d t h e b o u n d a r y y t h a t b e l o n g s t o t h e 1,

136

Moscow. Translated from Izvestiya pp. 175-179, January-February, 1979.

0015-4628/79/1401-0136507.50

A k a d e m i i Nauk SSSR, M e k h a n i k a Z h i d k o s t i i Gaza, Original article s u b m i t t e d December 12, 1977.

9 1979 Plenum Publishing

Corporation

No.

J #

a

J 5

J 5 3'

#

#

# Fig.

72

Z

i

o u t f l o w s e c t i o n (before the w a t e r r e a c h e s the outflow, F and F 0 c o i n c i d e ) . y s e p a r a t e s the w a t e r f r o m the oil on the o t h e r side of the outflow.

The b o u n d a r y

B e f o r e the a r r i v a l of the w a t e r at the outflow, the m o d e l of flow of two immiscible, i n c o m p r e s s i b l e f l u i d s p r o p o s e d in [{] is d e s c r i b e d by the f o l l o w i n g s y s t e m of e q u a t i o n s and b o u n d a r y c o n d i t i o n s on the interface: k V~ = - - - V p .

div Vi=O,

Pl-Pi=P~,

and t h e r e are also the b o u n d a r y

i=1,2

pk=ofmk -i

k Opt

k Op2

p,i

~t2

On

and initial

(1.1)

(i. 2)

(1.3)

On

conditions~

Here, k is the p e r m e a b i l i t y field; Pi, Vi, and ~i are the p r e s s u r e , velocity, and v i s c o s i t y of f l u i d i; g is the i n t e r p h a s e tension; m is the porosity; and n is the n o r m a l to F 0. The i n d i c e s i = l, 2 r e f e r to the w a t e r and oil, r e s p e c t i v e l y . The d i s p l a c e m e n t of the i n t e r f a c e tion with initial condition

is d e t e r m i n e d by s o l v i n g

the l i n e a r

transport

equa-

~Z

m--+

Ot

and corresponding fore,

V=Vt, (z,y)mDi;

VVz=0, boundary

V~, (x,y) E DI,

U s i n g the f u n c t i o n z, we can w r i t e Eqs. c o m p l e t e r e g i o n D in the f o r m

V=-gVp,

to d e t e r m i n e

(1.1)

g=k[z~i-i+(i-z)~2-~],

p(0, y , t ) = p _ ,

p(l,y,t)=p+,

the r e g i o n s D l and D 2 and,

and the b o u n d a r y c o n d i t i o n s

p=p~, (x, y)ED~; Ap=p_-p+~0

v~(x, ~ t)=v~(z, ~, t)=0 Once t h e w a t e r r e a c h e s t h e o u t f l o w , t h e b o u n d a r i e s and therefore on the boundary y it is necessary to take o u t f l o w of the w e t t i n g

O, (x, y)mD2

(1.4)

conditions.

The s o l u t i o n of (1.4) m a k e s it p o s s i b l e the p o s i t i o n of the i n t e r f a c e .

divV=O,

z(x, y, 0)=1, (x, y) EDt;

Pl, (x, y)~Dz

there-

for the

(1.5) (1.6) (1.7)

F and F a r e no l o n g e r i d e n t i c a l , 0 into account the condition of

fluid.

F o r this, we c o n s i d e r the p r o b l e m of d i s p l a c e m e n t of oil by w a t e r from a c i r c u l a r s t r a i g h t h y d r o p h i l i e c a p i l l a r y of r a d i u s r. The i n t e r f a c e b e t w e e n the two liquids is

i37

TABLE J

1

I=I{ 2.

7 56

X~ ll~

4 3 2 1

2 2 0

1 0 --s 0 0 4 --2

4

3

I -

--3 O I 2

2 3 2 I

2

5

0 1 0 --2 O 0 2

6

8

7

9

J

l

I --2

0

A l

io

i3

--2

0 0 --2

--2

~2

il

2

i&

O 1

2 --4

3

--1

0 0

1 0 1

--1 3

I

2

0

2

0

c o n c a v e to the r e g i o n o c c u p i e d by the water, and n e a r the i n t e r f a c e the p r e s s u r e in the w a t e r is less than the p r e s s u r e in the oil. If we i g n o r e the d e p e n d e n c e of the m e n i s c u s p r o f i l e on the a p p l i e d h y d r o d y n a m i c d i f f e r e n c e of the p r e s s u r e p, then the m e n i s c u s will h a v e the f o r m of a s p h e r i c a l s u r f a c e of r a d i u s r and the d i f f e r e n c e of the p r e s s u r e s across the i n t e r f a c e w i l l be Pk = 2~ W h e n the m e n i s c u s r e a c h e s the o u t f l o w section, the m o t i o n of the part of the interface that t o u c h e s the w a l l s of the c a p i l l a r y stops, and the b o u n d a r y comes to have a flat p r o f i l e w i t h Pk d e c r e a s i n g to zero. U n d e r the i n f l u e n c e of the a p p l i e d p r e s s u r e d i f f e r e n c e , the m e n i s c u s is " t u r n e d over" and the p r e s s u r e in the w a t e r b e c o m e s g r e a t e r than the p r e s sure in the oil. O n c e the p r e s s u r e d i f f e r e n c e is g r e a t e r than 2or -I, the w a t e r flows out of the c a p i l l a r y . If the p r e s s u r e d i f f e r e n c e Ap is i n s u f f i c i e n t to p r o d u c e s u c h a difference, the m e n i s c u s a c q u i r e s a f o r m for w h i c h the c a p i l l a r y forces are c o m p e n s a t e d by the h y d r o d y n a m i c p r e s s u r e d i f f e r e n c e , and the m e n i s c u s stops. One can show that the time r e q u i r e d for the b r e a k t h r o u g h of the w a t e r that has r e a c h e d the o u t f l o w is short c o m p a r e d w i t h the t i m e of d i s p l a c e m e n t of the oil f r o m the c a p i l l a r y . The simulated flow becomes m u c h m o r e c o m p l i c a t e d w h e n one c o n s i d e r s d i s p l a c e m e n t f r o m a m e s h of c a p i l l a r i e s of arb i t r a r y radius. B e a r i n g in m i n d this n a t u r e of the f l o w of the w a t e r out of the h y d r o p h i l i c capillary, we shall a s s u m e that the w a t e r b r e a k s t h r o u g h on the p a r t of the b o u n d a r y y at w h i c h the condition p+-p~=-p~ (1.8) is

satisfied.

We c a l l this part of y the region of outflow of the water. After the water has b r o k e n through, the p r e s s u r e in this r e g i o n of the o u t f l o w s e c t i o n is c o n t i n u o u s , i.e., the c o n d i t i o n (1.8) no l o n g e r holds.

The w e t t i n g p h a s e d o e s not b r e a c h the r e m a i n i n g p a r t of TR e m e m b e r i n g that the outflow s e c t i o n is an i s o b a r on the outside, and that oil c a n n o t f l o w into the stratum, the v e l o c i t y on this p a r t of 7 is zero (V 2 = 0). T h e n f r o m (1.3),

Vlx=O Thus,

the c o n d i t i o n s

(1.9)

s a t i s f i e d on y d e p e n d on the state of the system.

N o t e that the a n a l o g y b e t w e e n the c o n d i t i o n s (1.8) and (1.9) on the o u t f l o w and the c o n d i t i o n of flow of w a t e r out of the c a p i l l a r y has b e e n m a d e u n d e r the a s s u m p t i o n that the t i m e r e q u i r e d for the " t u r n o v e r " of the m e n i s c u s is short, i.e., i g n o r i n g the p r o c e s s of d e f o r m a t i o n o f the m e n i s c u s and the r e l a t e d n a t u r e of the c h a n g e of the c a p i l l a r y pressure, we a s s u m e that at any point of y the w a t e r is e i t h e r at rest or b r e a c h e s 7 i n s t a n taneously. The s y s t e m (1.2)-(1.9) w i t h c o r r e s p o n d i n g b o u n d a r y c o n d i t i o n for z at the i n f l o w s i m u l a t e s the c o m p l e t e p r o c e s s of d i s p l a c e m e n t of the oil by the water. T h e p r o b l e m of the m o t i o n of the i n t e r f a c e was s o l v e d by a f i n i t e - d i f f e r e n c e method. To find the s o l u t i o n of the linear e q u a t i o n (1.4), we u s e d the f i n i t e - d i f f e r e n c e " c o r n e r " scheme. S i n c e this m e t h o d leads to a s t r o n g " s m e a r i n g " of the n u m e r i c a l solution, the f u n c t i o n z of Eq. (1.4) was d e t e r m i n e d by the n u m e r i c a l s o l u t i o n of a s p e c i a l l y c o n s t r u c t e d q u a s i l i n e a r t r a n s p o r t e q u a t i o n [4]. To d e t e r m i n e the p r e s s u r e field, Eqs. (i.5) w i t h a l l o w a n c e for (1.2), (1.3), and (1.8) w e r e a p p r o x i m a t e d u s i n g the i n t e g r o - i n t e r p o l a t i o n

138

0,8

o.B x/~

~/~

I

Fig. m e t h o d of

1

a

2

[5] b y . a c o n s e r v a t i v e

b

c

Fig. difference

d

e

3

s c h e m e of s e c o n d o r d e r of a c c u r a c y

2. S u p p o s e that at the i n i t i a l time the flow r e g i o n is f i l l e d w i t h oil and that w a t e r is s u p p l i e d at the i n f l o w s e c t i o n (z(0, y, t) = i).

[2, 4].

(z(x,

y~ 0) = 0)

F i g u r e 1 shows the p o s i t i o n s of the i n t e r f a c e for concurrent-countercurrent p r o b l e m s of i m h i b i t i o n (a) and d i s p l a c e m e n t w i t h Ap = 0.2 arm (b) and Ap = 2 atm (c). The c a l c u l a tions w e r e m a d e for the f o l l o w i n g data: 1 = 20 cm, b = 5 cm, ~ = 20 dyn/cm, ~I = i cP, ~2 = 5 cP, m = 0.2. T a b l e 1 g i v e s the f i e l d of I00 (k -- l) f l u c t u a t i o n s of the p e r m e a b i l i t y in I and J b l o c k s [ R u s s i a n o b s c u r e ] . The m e a n v a l u e of the p e r m e a b i l i t y is 1 darcy, and the rms d e v i a t i o n 0.02 darcy. We u s e d a 42 • 21 mesh. C u r v e s 1 and 2 are d r a w n for t i m e s c o r r e s p o n d i n g to ~ = 20 and 40% oil e x t r a c t i o n ; c u r v e s 3 c o r r e s p o n d to the times of a r r i v a l of the w a t e r at the outflow, h a v i n g the v a l u e s 63.7, 65, and 71% for the p r o b l e m s a, b, and c, r e s p e c t i v e l y . The f u r t h e r m o t i o n of the i n t e r f a c e is r e l a t e d to f u l f i l l m e n t of the c o n d i t i o n s (1.8) and (1.9). It is o b v i o u s that for i m b i b i t i o n (Ap = 0) the c o n d i t i o n (I.8) c a n n o t be satisfied. T h e c o n d i t i o n of no o u t f l o w of w a t e r has the c o n s e q u e n c e that the w a t e r fills the c o m p l e t e o u t f l o w s e c t i o n (Fig. la, c u r v e 4). 2 4 . 5 % of the oil (the h a t c h e d region) r e m a i n s in the stratum. It is n a t u r a l to e x p e c t that for d i s p l a c e m e n t the n a t u r e of the o u t f l o w of the w a t e r w i l l d e p e n d on the a p p l i e d h y d r o d y n a m i c p r e s s u r e d i f f e r e n c e Ap. It is o b v i o u s that for Ap<6, w h e r e ~=min pk(l, y), the w a t e r will not f l o w t h r o u g h the o u t f l o w section. Let us O~i~b

consider

flow with.hp>8.

As can be s e e n from Fig. Ib, the water, w h i c h a r r i v e s at the o u t f l o w in the f o r m of a "tongue'*, f l o w s t h e n a l o n g the o u t f l o w section, the p r e s s u r e in the w a t e r n e a r the outflow i n c r e a s i n g . As s o o n as the w a t e r p r e s s u r e at some p o i n t of the o u t f l o w s e c t i o n e x c e e d s the b o u n d a r y p r e s s u r e by the a m o u n t of the c a p i l l a r y jump, the w a t e r flows out of the s a m p l e (curve 4; the o u t f l o w r e g i o n of the w a t e r is b o u n d e d by the arrows). Note that at the t i m e of breakthrough and a f t e r it the c o m p l e t e o u t f l o w s e c t i o n is not f i l l e d w i t h water, i.e., one o b s e r v e s the concurrent m o t i o n of oil and water. In addition, the r e g i o n s of o u t f l o w of the w a t e r and of its a c c u m u l a t i o n at the o u t f l o w s e c t i o n are different. The times of a r r i v a l and b r e a k t h r o u g h of the w a t e r also d i f f e r c o n s i d e r a b l y . At the time of b r e a k t h r o u g h , 2 0 . 5 % of the oil r e m a i n s in the s t r a t u m (the h a t c h e d r e g i o n in Fig. ib), i.e., 12.5% of the oil is o b t a i n e d d u r i n g the w a t e r l e s s p e r i o d after the a r r i v a l of the w a t e r at the o u t f l o w section. The d e s c r i b e d e x a m p l e shows that w h e n real p r e s s u r e g r a d i e n t s are s p e c i f i e d a l l o w a n c e for the end e f f e c t has a s i g n i f i c a n t i n f l u e n c e on the d i s p l a c e m e n t of the i n t e r f a c e . At large g r a d i e n t s , the i n f l u e n c e d e c r e a s e s . It can be s e e n f r o m Fig. Ic that for the p ~ o b l e m w i t h Ap = 2 atm the w a t e r b r e a k s t h r o u g h i m m e d i a t e l y at the p o i n t at w h i c h the w a t e r r e a c h e s the o u t f l o w section. C u r v e 4 c o r r e s p o n d s to the v a l u e ~ = 85%, w i t h p u m p i n g t h r o u g h of 88% of the p o r e w a t e r volume. Thus, the f l o w g e o m e t r y and e x t e n t of d i s p l a c e m e n t a f t e r the w a t e r has r e a c h e d the o u t f l o w d e p e n d s t r o n g l y on the flow v e l o c i t y . In Figs. 2a-2c, we show the d i s t r i b u t i o n s a l o n g the x axis of the f r a c t i o n s of the w e t t i n g p h a s e in the t r a n s v e r s e s e c t i o n corr e s p o n d i n g to the p r o b l e m s a)-c). C u r v e s 1 and 2 c o r r e s p o n d to the p o s i t i o n s of the i n t e r f a c e 3 and 4 in Fig. 1. N o t e the q u a l i t a t i v e a g r e e m e n t in the n a t u r e of the flow b e t w e e n the e x p e r i m e n t a l and t h e o r e t i c a l i n v e s t i g a t i o n s . To treat the end effect in the R a p o p o r t - - L e a s m o d e l the b r e a k t h r o u g h c o n d i t i o n of the w e t t i n g p h a s e is u s u a l l y taken to be

139

independent of the velocity IS], which leads to a growthbin the water saturation near the outflow section for any velocity. Nevertheless, such a f o r m u l a t i o n correctly takes into account the tendency for the end effect to decrease when the displacement rate is increased. Let us consider in more detail the dynamics of the interface after the water has arrived at the outflow section for the example of the displacement problem with Ap = 0.2 arm (Fig. lb). Figure 3 shows the distributions of the phases for the part of the stratum adjoining the outflow section. The distributions a)-e) correspond to the values n = 76.1, 77.5, 79.4, 80.7, and 81%. The region D 2 is hatched. Since the hydrodynamic pressure gradients in the transverse direction are small, the water flows along the outflow section under the influence of the capillary forces. Simultaneously, a second "tongue" of water arrives at the outflow section, and this merges w i t h the water that is flowing along it. As a result, a block of oil is formed near the o u t f l o w section (a). Under the influence of the pressure gradient in the longitudinal direction, the oil, displacing the water, advances to the outflow (b). However, the capillary forces force the water to enter the area occupied by the oil, and a block of oil is again formed. This pulsating outflow of the oil can occur at different positions over the outflow section, and both before and after b r e a k t h r o u g h of the water (d, e). This phenomenon agrees with the experiments of [6], in which it was found that after the appearance of the water at the outflow the displaced phase does not come out in a continuous jet but in individual drops LITERATURE

CITED

I. J. G. Richardson, J. K. Kerver, J . A. Hafford, and J. S. Osoba, "Laboratory determination of relative permeability," Trans. AIME, 195 (1952). 2. P. V. Indel'man, R. M. Kats, and M. I. Shvidler, "A model of flow of immiscible liquids," Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 6 (1977). 3. R. E. Collins, Flow of Fluids Through Porous Materials, Reinhold, New York (1961). 4. P. V. Indel'man, P. V. Kats, and M. I. Schvidler, " I n v e s t i g a t i o n into unstable displacement by numerical simulation," in: Numerical Solution of Problems in the Flow of a M a n y - P h a s e Incompressible Fluid [in Russian], VTs SO AN SSSR, N o v o s i b i r s k (1977). 5. A. A. Samarskii, T h e o r y of Difference Schemes [in Russian], Nauka, Moscow (1977). 6. G. A. Babalyan, Questions of the M e c h a n i s m of Oil E x t r a c t i o n [in Russian], Aznefteizdat, Baku (1956).

RELATIVE P E R M E A B I L I T I E S

FOR THE LIQUID AND GAS FOR

FOAMING IN A POROUS M E D I U M UDC 5 3 2 . 5 4 6

M. F. Karimov and M. M. K h a r i s o v

We have studied experimentally the relative p e r m e a b i l i t i e s for liquid and gas for foaming in a porous medium. We have established that the gas p e r m e a b i l i t y depends on the saturation and absolute c o n c e n t r a t i o n of the foaming agent. We have made a statistical estimate of the c o r r e l a t i o n of the approximating p o w e r - l a w and polynomial functions. We compare the theoretical values of the front saturation obtained in the Buckley--Leverett scheme in conjunction w i t h the a p p r o x i m a t i n g function with the experimental values. The theory of the simultaneous flow in a porous m e d i u m of two-phase systems is based on the relative p e r m e a b i l i t i e s of the n o n i n t e r a c t i n g phases obtained in filtration. Modern methods of intensification of the mutual replacement of liquid and gas foresee physicochemical transformations in the porous medium. Under these conditions, the use in calculations of the w e l l - k n o w n dependences of Wyckoff and Botset leads to a large error; in particular, the difference between the front saturation determined in accordance with the Buckley--Leverett theory and the experimental value is several tens of percent. One of the effective methods of intensifying the displacement of a stratal liquid is to use f o a m - g e n e r a t i n g solutions. Generation of foam when the liquid is displaced by the gas leads to a stable m o t i o n of the phase interface and ensures virtually piston-like Ufa. T r a n s l a t e d from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. i, pp. 179-182, January-February, 1979. Original article submitted September 29, 1977.

140

0015-4628/79/1401-0140507.50

9 1979

Plenum Publishing

Corporation